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Theorem dn1dc 960
Description: DN1 for decidable propositions. Without the decidability conditions, DN1 can serve as a single axiom for Boolean algebra. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
dn1dc  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  ->  ( -.  ( -.  ( -.  ( ph  \/  ps )  \/  ch )  \/  -.  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )  <->  ch ) )

Proof of Theorem dn1dc
StepHypRef Expression
1 pm2.45 738 . . . . 5  |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
2 imnan 690 . . . . 5  |-  ( ( -.  ( ph  \/  ps )  ->  -.  ph ) 
<->  -.  ( -.  ( ph  \/  ps )  /\  ph ) )
31, 2mpbi 145 . . . 4  |-  -.  ( -.  ( ph  \/  ps )  /\  ph )
43biorfi 746 . . 3  |-  ( ch  <->  ( ch  \/  ( -.  ( ph  \/  ps )  /\  ph ) ) )
5 orcom 728 . . 3  |-  ( ( ch  \/  ( -.  ( ph  \/  ps )  /\  ph ) )  <-> 
( ( -.  ( ph  \/  ps )  /\  ph )  \/  ch )
)
6 ordir 817 . . 3  |-  ( ( ( -.  ( ph  \/  ps )  /\  ph )  \/  ch )  <->  ( ( -.  ( ph  \/  ps )  \/  ch )  /\  ( ph  \/  ch ) ) )
74, 5, 63bitri 206 . 2  |-  ( ch  <->  ( ( -.  ( ph  \/  ps )  \/  ch )  /\  ( ph  \/  ch ) ) )
8 pm4.45 784 . . . . . 6  |-  ( ch  <->  ( ch  /\  ( ch  \/  th ) ) )
9 simprrl 539 . . . . . . 7  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  -> DECID  ch )
10 dcor 935 . . . . . . . . 9  |-  (DECID  ch  ->  (DECID  th 
-> DECID  ( ch  \/  th )
) )
1110imp 124 . . . . . . . 8  |-  ( (DECID  ch 
/\ DECID  th )  -> DECID  ( ch  \/  th ) )
1211ad2antll 491 . . . . . . 7  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  -> DECID  ( ch  \/  th ) )
13 anordc 956 . . . . . . 7  |-  (DECID  ch  ->  (DECID  ( ch  \/  th )  ->  ( ( ch  /\  ( ch  \/  th )
)  <->  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) ) )
149, 12, 13sylc 62 . . . . . 6  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  ->  (
( ch  /\  ( ch  \/  th ) )  <->  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )
158, 14bitrid 192 . . . . 5  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  ->  ( ch 
<->  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )
1615orbi2d 790 . . . 4  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  ->  (
( ph  \/  ch ) 
<->  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) ) )
1716anbi2d 464 . . 3  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  ->  (
( ( -.  ( ph  \/  ps )  \/ 
ch )  /\  ( ph  \/  ch ) )  <-> 
( ( -.  ( ph  \/  ps )  \/ 
ch )  /\  ( ph  \/  -.  ( -. 
ch  \/  -.  ( ch  \/  th ) ) ) ) ) )
18 dcor 935 . . . . . . . 8  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  \/  ps )
) )
19 dcn 842 . . . . . . . 8  |-  (DECID  ( ph  \/  ps )  -> DECID  -.  ( ph  \/  ps ) )
2018, 19syl6 33 . . . . . . 7  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  -.  ( ph  \/  ps ) ) )
2120imp 124 . . . . . 6  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID  -.  ( ph  \/  ps ) )
2221adantrr 479 . . . . 5  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  -> DECID  -.  ( ph  \/  ps ) )
23 dcor 935 . . . . 5  |-  (DECID  -.  ( ph  \/  ps )  -> 
(DECID 
ch  -> DECID 
( -.  ( ph  \/  ps )  \/  ch ) ) )
2422, 9, 23sylc 62 . . . 4  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  -> DECID  ( -.  ( ph  \/  ps )  \/  ch ) )
25 dcn 842 . . . . . . . 8  |-  (DECID  ch  -> DECID  -.  ch )
269, 25syl 14 . . . . . . 7  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  -> DECID  -.  ch )
27 dcn 842 . . . . . . . 8  |-  (DECID  ( ch  \/  th )  -> DECID  -.  ( ch  \/  th )
)
2812, 27syl 14 . . . . . . 7  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  -> DECID  -.  ( ch  \/  th ) )
29 dcor 935 . . . . . . 7  |-  (DECID  -.  ch  ->  (DECID  -.  ( ch  \/  th )  -> DECID  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )
3026, 28, 29sylc 62 . . . . . 6  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  -> DECID  ( -.  ch  \/  -.  ( ch  \/  th ) ) )
31 dcn 842 . . . . . 6  |-  (DECID  ( -. 
ch  \/  -.  ( ch  \/  th ) )  -> DECID  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) )
3230, 31syl 14 . . . . 5  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  -> DECID  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) )
33 dcor 935 . . . . . 6  |-  (DECID  ph  ->  (DECID  -.  ( -.  ch  \/  -.  ( ch  \/  th ) )  -> DECID  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) ) )
3433imp 124 . . . . 5  |-  ( (DECID  ph  /\ DECID  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) )  -> DECID  ( ph  \/  -.  ( -. 
ch  \/  -.  ( ch  \/  th ) ) ) )
3532, 34syldan 282 . . . 4  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  -> DECID  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )
36 anordc 956 . . . 4  |-  (DECID  ( -.  ( ph  \/  ps )  \/  ch )  ->  (DECID  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) )  -> 
( ( ( -.  ( ph  \/  ps )  \/  ch )  /\  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )  <->  -.  ( -.  ( -.  ( ph  \/  ps )  \/  ch )  \/  -.  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) ) ) ) )
3724, 35, 36sylc 62 . . 3  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  ->  (
( ( -.  ( ph  \/  ps )  \/ 
ch )  /\  ( ph  \/  -.  ( -. 
ch  \/  -.  ( ch  \/  th ) ) ) )  <->  -.  ( -.  ( -.  ( ph  \/  ps )  \/  ch )  \/  -.  ( ph  \/  -.  ( -. 
ch  \/  -.  ( ch  \/  th ) ) ) ) ) )
3817, 37bitrd 188 . 2  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  ->  (
( ( -.  ( ph  \/  ps )  \/ 
ch )  /\  ( ph  \/  ch ) )  <->  -.  ( -.  ( -.  ( ph  \/  ps )  \/  ch )  \/  -.  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) ) ) )
397, 38bitr2id 193 1  |-  ( (DECID  ph  /\  (DECID  ps  /\  (DECID  ch  /\ DECID  th )
) )  ->  ( -.  ( -.  ( -.  ( ph  \/  ps )  \/  ch )  \/  -.  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )  <->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835
This theorem is referenced by: (None)
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